Abstract
Consider the heat kernel on the universal cover of a closed Riemannian manifold of negative sectional curvature. We show the local limit theorem for ℘:
where is the bottom of the spectrum of the geometric Laplacian and is a positive -harmonic function which depends on . We also show that the -Martin boundary of is equal to its topological boundary. The Martin decomposition of gives a family of measures on . We show that is a family minimizing the energy or Mohsen’s Rayleigh quotient. We apply the uniform Harnack inequality on the boundary and the uniform three-mixing of the geodesic flow on the unit tangent bundle for suitable Gibbs–Margulis measures.
Citation
François Ledrappier. Seonhee Lim. "Local limit theorem in negative curvature." Duke Math. J. 170 (8) 1585 - 1681, 1 June 2021. https://doi.org/10.1215/00127094-2020-0069
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